Mathematical Sciences: Topics in Probability Theory: Statistical Mechanics Type Models and Sums of Independent Random Variables

概率论中有几个问题需要研究。前三个模型与统计物理密切相关，而后一个模型则涉及更传统的主题，即独立的、同分布的随机变量的和。具体的问题包括寻找随机增长模型半径的界，研究随机电阻网络的一些性质，以及研究第一次和最后一次通过的渗流的渐近形状。这项研究将调查一些受统计物理启发的概率模型。这些包括渗流理论中的临界现象、扩散限制聚集和首次通过渗流。此外，研究人员还将研究独立的、同分布的随机变量和发散的标准，其中一些大的求和已被去掉。

Several problems in probability theory are to be studied. The first three concern models which are closely related to statistical physics, while the last one deals with the more traditional topic of sums of independent, identically distributed random variables. Particular problems include finding bounds on the radius of a stochastic growth model, investigating some properties of random resistor networks, and studying the asymptotic shape of first and last passage percolation. This research will investigate some probabilistic models inspired by statistical physics. These include critical phenomena in percolation theory, diffusion limited aggregation , and first-passage percolation. In addition, the investigator will also look at criteria for divergence of sums of independent, identically distributed random variables from which some large summands have been removed.